In recent years, one of the most interesting developments in quantum mechanics has
been the construction of new exactly solvable potentials connected with the appearance
of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In
contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials,
which start with a constant, the EOP families begin with some polynomial of degree
greater than or equal to one, but still form complete, orthogonal sets with respect
to some positive-definite measure. We show how they may appear in the bound-state
wavefunctions of some rational extensions of well-known exactly solvable quantum potentials.
Such rational extensions are most easily constructed in the framework of supersymmetric
quantum mechanics (SUSYQM), where they give rise to a new class of translationally
shape invariant potentials. We review the most recent results in this field, which
use higher-order SUSYQM. We also comment on some recent re-examinations of the shape
invariance condition, which are independent of the EOP construction problem.